Implications

This non-implication, Form 189 \( \not \Rightarrow \) Form 115, whose code is 6, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9615, whose string of implications is:
    191 \(\Rightarrow\) 189
  • A proven non-implication whose code is 5. In this case, it's Code 3: 475, Form 191 \( \not \Rightarrow \) Form 119 whose summary information is:
    Hypothesis Statement
    Form 191 <p> \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). </p>

    Conclusion Statement
    Form 119 <p> <strong>van Douwen's choice principle:</strong> \(C(\aleph_{0}\),uniformly orderable with order type of the integers): Suppose \(\{ A_{i}: i\in\omega\}\) is a set and there is a function \(f\) such that for each \(i\in\omega,\ f(i)\) is an ordering of \(A_{i}\) of type \(\omega^{*}+\omega\) (the usual ordering of the integers), then \(\{A_{i}: i\in\omega\}\) has a choice function. </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 7445, whose string of implications is:
    115 \(\Rightarrow\) 118 \(\Rightarrow\) 119

The conclusion Form 189 \( \not \Rightarrow \) Form 115 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N2(\hbox{LO})\) van Douwen's Model This model is another variationof \(\cal N2\)

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